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Properties

Label 2-552-69.68-c1-0-14
Degree $2$
Conductor $552$
Sign $0.744 + 0.667i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.47 − 0.909i)3-s + 3.63·5-s − 1.67i·7-s + (1.34 + 2.68i)9-s + 0.850·11-s + 3.40·13-s + (−5.35 − 3.30i)15-s − 3.63·17-s + 4.92i·19-s + (−1.52 + 2.47i)21-s + (0.850 − 4.71i)23-s + 8.18·25-s + (0.456 − 5.17i)27-s − 1.36i·29-s + 6.94·31-s + ⋯
L(s)  = 1  + (−0.851 − 0.525i)3-s + 1.62·5-s − 0.634i·7-s + (0.448 + 0.893i)9-s + 0.256·11-s + 0.944·13-s + (−1.38 − 0.852i)15-s − 0.880·17-s + 1.13i·19-s + (−0.333 + 0.539i)21-s + (0.177 − 0.984i)23-s + 1.63·25-s + (0.0878 − 0.996i)27-s − 0.253i·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.744 + 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41265 - 0.540748i\)
\(L(\frac12)\) \(\approx\) \(1.41265 - 0.540748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.47 + 0.909i)T \)
23 \( 1 + (-0.850 + 4.71i)T \)
good5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 + 1.67iT - 7T^{2} \)
11 \( 1 - 0.850T + 11T^{2} \)
13 \( 1 - 3.40T + 13T^{2} \)
17 \( 1 + 3.63T + 17T^{2} \)
19 \( 1 - 4.92iT - 19T^{2} \)
29 \( 1 + 1.36iT - 29T^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
37 \( 1 + 10.5iT - 37T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 - 0.634iT - 43T^{2} \)
47 \( 1 + 0.275iT - 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 3.74iT - 59T^{2} \)
61 \( 1 + 7.64iT - 61T^{2} \)
67 \( 1 - 8.17iT - 67T^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 + 5.86T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 18.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.60906152395280497109899202197, −10.06779184928713646269388282249, −9.023616471993520121012019710734, −7.935200918536033112391687925422, −6.61432255475141257413313328737, −6.27237480303246102464734534690, −5.34151913272110397613004026358, −4.16805954900663586029148186332, −2.29260461430445556177432491355, −1.18017307149640458402427395063, 1.46436778415693328027134450368, 2.93186082143668215986145946541, 4.54822760989082436603039872454, 5.43611717130628844992035563057, 6.20770171189199719630958668124, 6.78931942269978127229295223263, 8.632487236191707507847541662778, 9.303866524882358603313212522122, 9.956399264618167963952339631061, 10.89350175779470110184426744526

Graph of the $Z$-function along the critical line